Summary: We analyze the stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist.
First we show that the disease-free equilibrium is globally asymptotically stable if and only if . Second we show that the model is permanent if and only if . Moreover, using a threshold parameter characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for and it loses stability as the length of the delay increases past a critical value for . Our result is an extension of the stability results of J.-J. Wang, J.-Z. Zhang and Z. Jin [Analysis of an SIR model with bilinear incidence rate. Nonlinear Anal., Real World Appl. 11, No. 4, 2390–2402 (2010; Zbl 1203.34136)].